The paper presents a geometric investigation of collision-free 3-axis milling of surfaces. We consider surfaces with a global shape condition: they shall be interpretable as the graphs of bivariate functions or shall be star-shaped with respect to a point. If those surfaces satisfy a local millability criterion involving curvature information, it is proved that this implies globally gouge-free milling. The proofs are based on general offset surfaces. The results can be applied to tool-motion planning and the computation of optimal cutter shapes.

1.
Blaschke, W., 1956, Kreis und Kugel, De Gruyter.
2.
Brechner, E., 1992, “General Offset Curves and Surfaces,” in Geometry Processing for Design and Manufacturing, ed. Barnhill, R. E., SIAM, Philadelphia, pp. 101–121.
3.
Chen
Y. J.
, and
Ravani
B.
,
1987
, “
Offset Surface Generation and Contouring
,”
J. Mech. Trans. Auto. Des.
, Vol.
109
, pp.
133
142
.
4.
Choi
B. K.
,
Kim
D. H.
, and
Jerard
R. B.
,
1997
, “
C-Space Approach to Tool-Path Generation for Die and Mould Machining
,”
Computer Aided Design
, Vol.
29
, No.
9
, pp.
657
669
.
5.
Dieudonne´ J., Treatise on Analysis, (8 Volumes) Academic Press, New York, 1960–1993.
6.
do Carmo, M. P., 1976, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NY.
7.
Glaeser, G., and Stachel, H., 1998, Open Geometry, Springer, New York.
8.
Glaeser, G., Wallner, J., and Pottmann, H., 1998, “Collision-Free 3-Axis Milling and Selection of Cutting-Tools,” to appear in Comp. Aided Design.
9.
Hirsch, M. W., 1976, Differential Topology, Graduate Texts in Mathematics 33, Springer.
10.
Hoschek, J., and Lasser, D., 1993, Fundamentals of Computer Aided Geometric Design, A. K. Peters, Wellesley, MA.
11.
Lee, I. K., Kim, M. S., and Elber, G., 1998, “Polynomial/Rational Approximation of Minkowski Sum Boundary Curves,” Graphical Models and Image Processing, to appear.
12.
Marciniak, K., 1991, Geometric Modelling for Numerically Controlled Machining, Oxford University Press, New York.
13.
Milnor, J., 1965, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville.
14.
Pottmann
H.
,
1997
, “
General Offset Surfaces
,”
Neural, Parallel & Scientific Computations
, Vol.
5
, pp.
55
80
.
15.
Sachs, H., 1990, Isotrope Geometrie des Raumes, Vieweg.
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